from isothermic triangulated surfaces to discrete ...lam/slides/oberwolfach_ddg2015.pdffrom...
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From isothermic triangulated surfaces todiscrete holomorphicity
Wai Yeung Lam
TU Berlin
Oberwolfach, 2 March 2015
Joint work with Ulrich Pinkall
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 1 / 33
Table of Content
1 Isothermic triangulated surfaces
Discrete conformality: circle patterns, conformal equivalence
2 Discrete minimal surfaces
Weierstrass representation theorem
3 Discrete holomorphicity
Planar triangular meshes
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 2 / 33
Isothermic Surfaces in the Smooth Theory
Surfaces in Euclidean space R3.
1 Definition: Isothermic if there exists a conformal curvature line parametrization.
2 Examples: surfaces of revolution, quadrics, constant mean curvature surfaces,
minimal surfaces.
3 Related to integrable systems.
Enneper’s Minimal Surface
Aim: a discrete analogue without conformal curvature line parametrizations.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 3 / 33
Isothermic Surfaces in the Smooth Theory
Surfaces in Euclidean space R3.
1 Definition: Isothermic if there exists a conformal curvature line parametrization.
2 Examples: surfaces of revolution, quadrics, constant mean curvature surfaces,
minimal surfaces.
3 Related to integrable systems.
Enneper’s Minimal Surface
Aim: a discrete analogue without conformal curvature line parametrizations.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 3 / 33
Isothermic Surfaces in the Smooth Theory
TheoremA surface in Euclidean space is isothermic if and only if locally there exists a non-trivial
infinitesimal isometric deformation preserving the mean curvature.
Cieslinski, Goldstein, Sym (1995)
Discrete analogues of
1 infinitesimal isometric deformations and
2 mean curvature
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 4 / 33
Isothermic Surfaces in the Smooth Theory
TheoremA surface in Euclidean space is isothermic if and only if locally there exists a non-trivial
infinitesimal isometric deformation preserving the mean curvature.
Cieslinski, Goldstein, Sym (1995)
Discrete analogues of
1 infinitesimal isometric deformations and
2 mean curvature
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 4 / 33
Triangulated SurfacesGiven a triangulated surface f : M = (V , E, F)→ R3, we can measure
1 edge lengths ` : E → R,
2 dihedral angles of neighboring triangles α : E → R and
3 deform it by moving the vertices.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 5 / 33
Infinitesimal isometric deformations
Definition
Given f : M → R3. An infinitesimal deformation f : V → R3 is isometric if ˙ ≡ 0.
If f isometric, on each face4ijk there exists Zijk ∈ R3 as angular velocity:
df(eij) = fj − fi = df(eij)× Zijk
df(ejk) = fk − fj = df(ejk)× Zijk
df(eki) = fi − fk = df(eki)× Zijk
If two triangles4ijk and4jil share a common edge eij , compatibility condition:
df(eij)× (Zijk − Zjil) = 0 ∀e ∈ E
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 6 / 33
Integrated mean curvature
A known discrete analogue of mean curvature H : E → R is defined by
He := αe`e.
But if ˙ = ˙H = 0 =⇒ α = 0 =⇒ trivial
Instead, we consider the integrated mean curvature around vertices H : V → R
Hvi :=∑
j
Heij =∑
j
αeij `ij .
If f preserves the integrated mean curvature additionally, it implies
0 = Hvi =∑
j
αij`ij =∑
j
〈df(eij), Zijk − Zjil〉 ∀vi ∈ V .
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 7 / 33
M∗ = combinatorial dual graph of M
∗e = dual edge of e.
Definition
A triangulated surface f : M → R3 is isothermic if there exists a R3-valued dual 1-form
τ such that ∑j
τ(∗eij) = 0 ∀vi ∈ V
df(e)× τ(∗e) = 0 ∀e ∈ E∑j
〈df(eij), τ(∗eij)〉 = 0 ∀vi ∈ V .
If additionally τ exact, i.e. ∃Z : F → R3 such that
Zijk − Zjil = τ(∗eij).
We call Z a Christoffel dual of f . Write f∗ := Z from now on...
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 8 / 33
The previous argument gives
CorollaryA simply connected triangulated surface is isothermic if and only if there exists a
non-trivial infinitesimal isometric deformation preserving H.
As in the smooth theory, we proved
TheoremThe class of isothermic triangulated surfaces is invariant under Möbius transformations.
We can transform τ explicitly under Möbius transformations
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 9 / 33
The previous argument gives
CorollaryA simply connected triangulated surface is isothermic if and only if there exists a
non-trivial infinitesimal isometric deformation preserving H.
As in the smooth theory, we proved
TheoremThe class of isothermic triangulated surfaces is invariant under Möbius transformations.
We can transform τ explicitly under Möbius transformations
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 9 / 33
Discrete conformality
Two notions of discrete conformality of a triangular mesh in R3:
1 circle patterns
2 conformal equivalence
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 10 / 33
Circle patterns
Circumscribed circles
Given f : M → R3, denote θ : E → (0, π] as the intersection angles of circumcircles.
Definition
We call f : V → R3 an infinitesimal pattern deformation if
θ ≡ 0
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Circumscribed circles Circumscribed spheres
TheoremA simply connected triangulated surface is isothermic if and only if there exists a
non-trivial infinitesimal pattern deformation preserving the intersection angles of
neighboring spheres.
Trivial deformations = Möbius deformations
Smooth theory: an infinitesimal conformal deformation preserving Hopf differential.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 12 / 33
Conformal equivalenceLuo(2004);Springborn,Schröder,Pinkall(2008);Bobenko et al.(2010)
i
j
kk
Definition
Given f : M → R3. We consider the length cross ratios lcr : E → R defined by
lcrij :=`jk`il
`ki`lj
Definition
An infinitesimal deformation f : V → R3 is called conformal if
˙lcr ≡ 0
Definition (Conformal equivalence of triangulatedsurfaces)
Two edge length functions `, ˜ : E → R are conformally equivalent if there exists
u : V → R such that˜
ij = eui+uj
2 `ij .
Definition
Given f : M → R3, an infinitesimal deformation f : V → R3 is conformal if there exists
u : V → R3 such that˙ij =
ui + uj
2`ij .
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Denote TfM = {infinitesimal conformal deformations of f}.
Theorem
For a closed genus-g triangulated surface f : M → R3, we have
dim TfM≥ |V | − 6g + 6.
The inequality is strict if and only if f is isothermic.
Smooth Theory: Isothermic surfaces are the singularities of the space of conformal
immersions.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 14 / 33
Example 1: Isothermic Quadrilateral Meshes
Definition (Bobenko and Pinkall, 1996)
A discrete isothermic net is a map f : Z2 → R3, for which all elementary quadrilaterals
have cross-ratios
q(fm,n, fm+1,n, fm+1,n+1, fm,n+1) = −1 ∀m, n ∈ Z,
Known: Existence of a mesh (Christoffel Dual) f∗ : Z2 → R3 such that for each quad
f∗m+1,n − f∗m,n = −fm+1,n − fm,n
||fm+1,n − fm,n||2
f∗m,n+1 − f∗m,n =fm,n+1 − fm,n
||fm,n+1 − fm,n||2
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Theorem
There exists an infinitesimal deformation f preserving the edge lengths and the
integrated mean curvature with
fm+1,n − fm,n = (fm+1,n − fm,n)× (f∗m+1,n + f∗m,n)/2,
fm,n+1 − fm,n = (fm,n+1 − fm,n)× (f∗m,n+1 + f∗m,n)/2.
Compared to the smooth theory:
df = df × f∗
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 16 / 33
Subdivision−−−−−−→
Theorem
There exists an infinitesimal deformation f preserving the edge lengths and the
integrated mean curvature with
fm+1,n − fm,n = (fm+1,n − fm,n)× (f∗m+1,n + f∗m,n)/2,
fm,n+1 − fm,n = (fm,n+1 − fm,n)× (f∗m,n+1 + f∗m,n)/2.
Compared to the smooth theory:
df = df × f∗
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 17 / 33
Subdivision−−−−−−→
Theorem
There exists an infinitesimal deformation f preserving the edge lengths and the
integrated mean curvature with
fm+1,n − fm,n = (fm+1,n − fm,n)× (f∗m+1,n + f∗m,n)/2,
fm,n+1 − fm,n = (fm,n+1 − fm,n)× (f∗m,n+1 + f∗m,n)/2.
Compared to the smooth theory:
df = df × f∗
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 18 / 33
Example 2: Homogeneous cyclinders
Pick g1, g2 ∈ Eucl(R3) which fix z-axis:
gi(p) =
cos θi sin θi 0
− sin θi cos θi 0
0 0 1
p +
0
0
hi
for some θi , hi ∈ R3. Note 〈g1, g2〉 ∼= Z2.
Together with an initial point p0 ∈ R3 gives
A strip of an isothermic triangulated cylinder
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 19 / 33
Example 3: Inscribed Triangulated Surfaces
Theorem
For a surface with vertices on a sphere, a R3-valued dual 1-form τ satisfying∑j
τ(∗eij) = 0 ∀vi ∈ V
df(e)× τ(∗e) = 0 ∀e ∈ E,
implies ∑j
〈df(eij), τ(∗eij)〉 = 0 ∀vi ∈ V .
CorollaryFor triangulated surfaces with vertices on a sphere, any infinitesimal deformation
preserving the edge lengths will preserve the integrated mean curvature.
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More examples of isothermic surfaces:
(a) Inscribed Triangular meshes with boundary (b) Jessen’s Orthogonal Icosahedron
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Table of Content
1 Isothermic triangulated surfaces
2 Discrete minimal surfaces
3 Discrete holomorphicity
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Discrete minimal surfacesSmooth theory: minimal surfaces are Christoffel duals of their Gauss images.
Definition
Given f : M → R3, a surface f∗ : M∗ → R3 is called a Christoffel dual of f if
df(e)× df∗(∗e) = 0 ∀e ∈ E, (1)∑j
〈df(eij), df∗(∗eij)〉 = 0 ∀vi ∈ V , (2)
Definition
f∗ : M∗ → R3 is called a discrete minimal surface if f : M → S2 is inscribed on the
unit sphere.
Note: if f is inscribed, then
(1) holds =⇒ (2) holds
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Equivalently,
discrete minimal surfaces = reciprocal-parallel meshes of inscribed triangulated surfaces
1 f∗ defined on dual vertices
2 dual edges parallel to primal edges
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Constructing discrete minimal surfacesEquivalent to find an infinitesimal rigid deformation of a planar triangular mesh
preserving the integrated mean curvature.
1 → a planar triangular mesh,
2 Infinitesimal rigid deformation of a planar triangular mesh: f = uN,
3 Preserving the integrated mean curvature =⇒∑
j(cotβ+ cot β)(uj − ui) = 0.
4 Inverse of stereographic projection
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 25 / 33
Weierstrass representation theorem
Recall in the smooth theory
Theorem
Given holomorphic functions f , h : U ⊂ C→ C such that f 2h is holomorphic. Then
f∗ : U → R3 defined by
df∗ = Re(
h(z)
f
(1− f 2)/2
(1 + f 2)/2
dz)
is a minimal surface.
In our setting : f(z) = z, h = 2iuzz where u : U → R is harmonic.
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Weierstrass representation theorem
Data: A planar triangular mesh f : M → R2 + a discrete harmonic function u : V → R.
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Table of Content
1 Isothermic triangulated surfaces
2 Discrete minimal surfaces
3 Discrete holomorphicity
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 28 / 33
Triangular meshes on CLuo(2004);Springborn,Schröder,Pinkall(2008);Bobenko et al.(2010)
Theorem
An infinitesimal deformation z : M → C is conformal if there exists u : V → R such
that˙|zj − zi | =
ui + uj
2|zj − zi |.
We call u the scaling factors.
Theorem
An infinitesimal deformation z : M → C is a pattern deformation if there exists
α : V → R such that
˙(
zj − zi
|zj − zi |) =
iαi + iαj
2
zj − zi
|zj − zi |.
We call iα the rotation factors.
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Theorem
An infinitesimal deformation z : V → C is conformal if and only if i z is a pattern
deformation.
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Theorem
Let z : M → C be an immersed triangular mesh and h : V → R be a function. The
following are equivalent.
1 h is a harmonic function∑j
(cotβk + cotβk)(hj − hi) = 0 ∀i ∈ V .
2 There exists pattern deformation i z with rotation factors ih. It is unique up to
infinitesimal scalings and translations.
3 There exists z conformal with scaling factors h. It is unique up to infinitesimal
rotations and translations.
(1) ⇐⇒ (2) in Bobenko, Mercat, Suris (2005)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 31 / 33
Pick a Möbius transformation φ : C→ C
z w := φ ◦ z
u harmonic ∃ u harmonic
f conformal dφ(f) conformal
φ
dφ
u unique up to a linear function.
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Thank you!
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